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Recent questions tagged determinant

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If $A$ and $B$ are square matrices of size $n\times n$, then which of the following statements is not true? $\det(AB)=\det(A) \det(B)$ $\det(kA)=k^n \det(A)$ $\det(A+B)=\det(A)+\det(B)$ $\det(A^T)=1/\det(A^{-1})$
asked Mar 31 in Linear Algebra Lakshman Patel RJIT 108 views
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Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is: $D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist.
asked Mar 31 in Linear Algebra Lakshman Patel RJIT 93 views
1 vote
1 answer
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The determinant $\begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix}$ equals $\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ $2\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ $3\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ None of these
asked Sep 23, 2019 in Linear Algebra Arjun 135 views
1 vote
1 answer
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Let $a$ be a non-zero real number. Define $f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$ for $x \in \mathbb{R}$. Then, the number of distinct real roots of $f(x) =0$ is $1$ $2$ $3$ $4$
asked Sep 23, 2019 in Linear Algebra Arjun 228 views
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2 answers
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The value of $\begin{vmatrix} 1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d \end{vmatrix}$ is $abcd(1+\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})$ $abcd(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})$ $1+\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}$ None of these
asked Sep 18, 2019 in Linear Algebra gatecse 179 views
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2 answers
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The value of $\begin{vmatrix} 1 & \log _x y & \log_x z \\ \log _y x & 1 & \log_y z \\ \log _z x & \log _z y & 1 \end{vmatrix}$ is $0$ $1$ $-1$ None of these
asked Sep 18, 2019 in Linear Algebra gatecse 78 views
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Let $A$ be an $n \times n$ matrix such that $\mid A^{2} \mid\: =1$. Here $\mid A \mid $ stands for determinant of matrix $A$. Then $\mid A \mid =1$ $\mid A \mid =0 \text{ or } 1$ $\mid A \mid =-1, 0 \text{ or } 1$ $\mid A \mid =-1 \text{ or } 1$
asked Sep 18, 2019 in Linear Algebra gatecse 96 views
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1 answer
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Let $A_{ij}$ denote the minors of an $n \times n$ matrix $A$. What is the relationship between $\mid A_{ij} \mid $ and $\mid A_{ji} \mid $? They are always equal $\mid A_{ij} \mid = – \mid A _{ji} \mid \text{ if } i \neq j$ They are equal if $A$ is a symmetric matrix If $\mid A_{ij} \mid =0$ then $\mid A_{ji} \mid =0$
asked Sep 18, 2019 in Linear Algebra gatecse 77 views
0 votes
1 answer
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The value of $\begin{vmatrix} 1+a& 1& 1& 1\\ 1&1+b &1 &1 \\ 1&1 &1+c &1 \\ 1&1 &1 &1+d \end{vmatrix}$ is $abcd(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$ $abcd(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$ $1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ None of these
asked Sep 18, 2019 in Linear Algebra gatecse 58 views
1 vote
1 answer
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The value of $\:\:\begin{vmatrix} 1&\log_{x}y &\log_{x}z \\ \log_{y}x &1 &\log_{y}z \\\log_{z}x & \log_{z}y&1 \end{vmatrix}\:\:$ is $0$ $1$ $-1$ None of these
asked Sep 18, 2019 in Linear Algebra gatecse 70 views
1 vote
1 answer
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Let $A$ be an $n\times n$ matrix such that $\mid\: A^{2}\mid=1.\:\: \mid A\:\mid$ stands for determinant of matrix $A.$ Then $\mid\:(A)\mid=1$ $\mid\:(A)\mid=0\:\text{or}\:1$ $\mid\:(A)\mid=-1,0\:\text{or}\:1$ $\mid\:(A)\mid=-1\:\text{or}\:1$
asked Sep 18, 2019 in Linear Algebra gatecse 76 views
1 vote
1 answer
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If $\begin{vmatrix} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{vmatrix} = k(10!)(11!)(12!)$, then the value of $k$ is $1$ $2$ $3$ $4$
asked Sep 18, 2019 in Linear Algebra gatecse 82 views
0 votes
1 answer
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If $f(x) = \begin{vmatrix} 2 \cos ^2 x & \sin 2x & – \sin x \\ \sin 2x & 2 \sin ^2 x & \cos x \\ \sin x & – \cos x & 0 \end{vmatrix},$ then $\int_0^{\frac{\pi}{2}} [ f(x) + f’(x)] dx$ is $\pi$ $\frac{\pi}{2}$ $0$ $1$
asked Sep 18, 2019 in Linear Algebra gatecse 115 views
0 votes
1 answer
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Let $A$ be a $3× 3$ real matrix with all diagonal entries equal to $0$. If $1 + i$ is an eigenvalue of $A$, the determinant of $A$ equals $-4$ $-2$ $2$ $4$
asked May 11, 2019 in Linear Algebra akash.dinkar12 436 views
0 votes
2 answers
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If $A =\begin{bmatrix} 2 &i \\ i & 0 \end{bmatrix}$ , the trace of $A^{10}$ is $2$ $2(1+i)$ $0$ $2^{10}$
asked May 11, 2019 in Linear Algebra akash.dinkar12 336 views
10 votes
6 answers
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Let $X$ be a square matrix. Consider the following two statements on $X$. $X$ is invertible Determinant of $X$ is non-zero Which one of the following is TRUE? I implies II; II does not imply I II implies I; I does not imply II I does not imply II; II does not imply I I and II are equivalent statements
asked Feb 7, 2019 in Linear Algebra Arjun 3.4k views
0 votes
0 answers
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If $\alpha, \beta$ and $\gamma$ are the roots of $x^3-px+q=0$, then the value of the determinant $\begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix}$ is $p$ $p^2$ $0$ $p^2+6q$
asked Sep 15, 2018 in Linear Algebra jothee 137 views
3 votes
2 answers
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If $\alpha, \beta$ and $\gamma$ are the roots of $x^3 - px +q = 0$, then the value of the determinant $\begin{vmatrix}\alpha & \beta & \gamma\\\beta & \gamma & \alpha\\\gamma & \alpha & \beta\end{vmatrix}$ is $p$ $p^2$ $0$ $p^2+6q$
asked Mar 28, 2018 in Linear Algebra jjayantamahata 419 views
16 votes
3 answers
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Let A be an mxn matrix and B an nxm matrix. It is given that determinant ( Im + AB ) = determinant ( In + BA ) , where Ik is the k×k identity matrix. Using the above property, the determinant of the matrix given below is $\begin{bmatrix} 2& 1& 1& 1\\ 1& 2& 1& 1\\ 1& 1& 2& 1\\ 1& 1& 1& 2 \end{bmatrix}$ A) 2 B) 5 C) 8 D) 16
asked Jun 21, 2017 in Linear Algebra the.brahmin.guy 2.2k views
7 votes
2 answers
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$\begin{vmatrix} 265 && 240 && 219 \\ 240 && 225 && 198 \\ 219 && 198 && 181 \\ \end{vmatrix}$ = 779 679 0 256
asked Jun 3, 2016 in Linear Algebra Desert_Warrior 2.1k views
6 votes
1 answer
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If the two matrices $\begin{bmatrix} 1 &0 &x \\ 0 & x& 1\\ 0 & 1 & x \end{bmatrix}$ and $\begin{bmatrix} x &1 &0 \\ x & 0& 1\\ 0 & x & 1 \end{bmatrix}$ have the same determinant, then the value of $x$ is $\frac{1}{2}$ $\sqrt2$ $\pm \frac{1}{2}$ $\pm \frac{1}{\sqrt2}$
asked Jun 1, 2016 in Linear Algebra jaiganeshcse94 1.3k views
19 votes
5 answers
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The determinant of the matrix given below is $\begin{bmatrix} 0 &1 &0 &2 \\ -1& 1& 1& 3\\ 0&0 &0 & 1\\ 1& -2& 0& 1 \end{bmatrix}$ $-1$ $0$ $1$ $2$
asked Nov 3, 2014 in Linear Algebra Ishrat Jahan 3k views
13 votes
1 answer
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The determinant of the matrix $\begin{bmatrix} 6 & -8 & 1 & 1 \\ 0 & 2 & 4 & 6 \\ 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & -1 \end{bmatrix}$ $11$ $-48$ $0$ $-24$
asked Sep 29, 2014 in Linear Algebra Kathleen 1.3k views
27 votes
6 answers
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If the matrix $A$ is such that $A= \begin{bmatrix} 2\\ −4\\7\end{bmatrix}\begin{bmatrix}1& 9& 5\end{bmatrix}$ then the determinant of $A$ is equal to ______.
asked Sep 28, 2014 in Linear Algebra jothee 3.7k views
23 votes
3 answers
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Which one of the following does NOT equal $\begin{vmatrix} 1 & x & x^{2}\\ 1& y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} \quad ?$ $\begin{vmatrix} 1& x(x+1)& x+1\\ 1& y(y+1) & y+1\\ 1& z(z+1) & z+1 \end{vmatrix}$ ... $\begin{vmatrix} 2& x+y & x^{2}+y^{2}\\ 2 & y+z & y^{2}+z^{2}\\ 1 & z & z^{2} \end{vmatrix}$
asked Sep 23, 2014 in Linear Algebra Arjun 3.1k views
15 votes
4 answers
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The determinant of the matrix $\begin{bmatrix}2 &0 &0 &0 \\ 8& 1& 7& 2\\ 2& 0&2 &0 \\ 9&0 & 6 & 1 \end{bmatrix}$ $4$ $0$ $15$ $20$
asked Sep 14, 2014 in Linear Algebra Kathleen 2k views
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